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@Danial Hast. This polynomial arose from a geometric problem regarding quadratic forms. The size of its fibres seem to be piecewise polynomial. Indeed, this is the first of a series of polynomials that arise essentially from the determinant of a Gram matrix.
@Nick the paper Alavi, Seyed Hassan; Burness, Timothy C. Large subgroups of simple groups. J. Algebra 421 (2015), 187–233 characterizes pairs $(G,H)$ with $G$ simple and $|H|\geqslant |G|^{1/3}$.
